Quantum Mechanics (QM) offers an incoherent explanation of microscopic world. They begin by telling you that the particles are actually waves as described by the so called Schrödinger Equation (SE). How can this be? They say the amplitude of the wave at a given reigion gives the probability of locating a point-like particle at that region. But almost immediately, you are told that SE is infact wrong because it is not compatible with relativity. So we need another equation termed 'Dirac Equation' (DE) to describe the waves. This DE in turn only performs a single cat walk show into the arena and is, in fact never used! We must immediately move on the Quantum Field Theory (QFT).

In QFT, we treat the wave described by DE not as a 'wave of probability' as the one in QM. Rather, breifly treat it like any other classic, like the acoustic waves. From there you quantize it. So it is called a 'second quantization'. What this means is that 'change the wave into an operator' (a meaningless mathematical statement which I will elaborate it Below).

Consider a wave on a rope. The height of the potions of the rope keeps changing as the section keep moving 'up and 'down'. So in the mathematical description of waves, we talk about 'change in height' as compared to 'change in time' and also as compared to 'change in distance' along the rope. So I am talking about how the height changes as I move along the rope, and as time goes. If I stay on the same spot, the height only changes with time and if I take a snapshot of the rope, the height only changes with distance. In the language of 'operators', (which QM loves so much), we split our description of the wave into two. In this case the 'height' and 'the change in'. So we are splitting the statement 'change in height' into 'change in' plus 'height'! the phrase 'change in' is called 'the opetator' and the phrase 'height' is callef 'statefunction'.

The QM snake oil peddler treats 'change in height' as a multiplication of two things: 'change in' times 'height'! They say the operator 'change in' is acting on a state function 'height' to bring about 'change in height' (think of how mulitplications of two values are present as to symbols sitting next to each other eg 2πr). So 'change in' is treated like π and 'heigh' is treated like r so that 'change in height' is a miltiplication of two values like πr, in that funny farm of quantum maths!

So that is what 'operator' means! We cange the DE from a decsription of a wave into a 'change in', leaving us with the question as to what corresponds to 'height' then in QFT. It turns out not be a wave at all. It is a sureal 'wave of waves'! Rather than thinking of a single wave wherein its amplitude gives the probability of locating a point-like particle, think of a monster 'wave of waves' whose 'amplitude' is the probability of finding a wave having a given characteristic!

So what were we looking for? The QM guy has told us in summary:

1.)QM paricles are waves described by SE
2.) SE is wrong and we must use DE
3.)DE is wrong still and we must use QFE
4.) In QFT, we don't describe the waves but actually some 'wave of waves'

So the question of what particles are is swept under the carpet in the supposedly correct QFT as it conveniently moves to the next question!!

DE (Dirac Equation) is also wrong! I will explain it this way. Consider the formula for kinetic energy of a moving object:

E=mv^2/2

This is Newtonian in the sense that it does not factor in relativistic mass increase, as we speed up the matter to attain velocity v. So the relativistic formula for kinetic energy is different. It is given by:

crucial point is that this relativistic formular reduces to the same Newt's formular when we make v very small so that squares of v are neglected in the Taylor series expansion of the square root. This is desired becase relativistic effects are not manifest in objects moving with small speeds.

But in DE, we find that we dont reproduce Schrodinger Equation when we make v very, very small! We reproduce an entirely different equation that still has the speed of light in it! So it is actually wrong to say that DE is a correction of SE to make it compatible with relativity. Dirac's theory is an entirely different theory that REPLACES Schrodinger's Theory! But they lie that it COMPLETES it!

The problem arises because Dirac did not quantize the above equation for the relativistic kinetic energy, in analogous way Schrodinger quantized the classic kinetic energy. Hence he fails to form a quantum theory which, like the classic theory, the relativistic quantum theory reduces to the classic quantum theory for small velocities. Instead Dirac opted to quantize the so called relativistic energy and momentum equation.

This equation reduces neither to the classic kinetic energy nor to the classic momentum when we make v 'small'. Instead it reduces to E=mc^2, as you can see by putting v=0. This is a relativistic theory, not a classic one of any sort. So this equation is wrong. But the detail errors done by the relativist in deriving this equation I will show you, perhaps another day.

To understand why the earlier equation is the right one for relativistic kinetic energy, begine by close examining how we derive the classic kinetic enery equation:

Then note that to do the analogous thing in the relativistic theory, we must factor in the fact that the mass increases from velocity to velocity. So we will not use a constant mass, m, but a relativistic mass, m', which is a function of velocity. Then we use the calculus technique of subtituting v/c=sinθ, and hence dv=cosθ, and use the Pythagorean realtionship: sin^2θ+cos^2θ=1 to remove the square root.

Then to understand how this equation reduces to the classic one for the kinetic energy, perform Taylor expansion  (or binomial expansion) of the square root and note that in first term of the expansion, mc^2 is canceled by subtraction and in the second term the c^2 is cancelled by division and we remain with the formula for the classic kinetic enery and the rest of the terms involves higher powers of v, which can be ignored if v is very small.

Schrodinger quantized the classic kinetic energy (in a process I will show you next). Therefore if Dirac quantized the above relativistic kinetic energy, he would have arrived at an equation that reduces to Schrodinger equation for small v. He quantized an equation that does not reproduces the kinetic energy. Therefore Dirac equation cannot reproduce Scrodinger equation for small velocities. Therefore the equations are contradictory when they are describing small velocities! This problem will stealthly translate to a problem in forming a relativistic model of 'point particle' which reduces to the classic quantum idea of 'probability of locating point particle'. So the theorists opted to just ignore the question and insist 'in quantum field theory, particles have no locations, only momentum'! It should not be! it should have 'something' that reduces to 'location' when the momentum is small. 

To quantize kinetic energy (hence arrive at the Schrodinger equation), use the debroglie hypothesis and assume that the quantum wave is a sinusoidal wave. These waves are described by the sin or cosine function.

Here, λ is the wavelength, h is Planck's Constant, E is energy and ω is the frequency. So we are also using Einstein's 'photoelectric law' to substitute for frequency. Next, we perform the second derivative of the sin function and note how kinetic energy jumps to the 'back'!

So the second derivative of  ψ forms what we call 'the kinetic part of Schrodinger equation'. So schrodinger equation can be thought of as an implicit way of stating the classic equation for energy having in mind that debroglie hypothesis relates momentum hence kinetic energy to waves via the wavelength relation. The full Schrodinger equation has the potential energy, V and it reads:

The second line is a way of stating it using the 'operator' language we saw. So the phrase:

Is called the 'Hamiltonian Operator', and as you can see, it is a meaningless statement! It says 'change in change in plus,...' without telling us what it is that is changing! What quantum guys did was to find a way in which such differential operators, when 'acting on state function', which simply means 'differenting the state functions and adding terms that contain multiplication with the state function' is analogous to multiplying an nxn square matrics with an nx1 matrics. So the nxn matrics is the operator while the nx1 matric is the state function. Since the nx1 matrics can also be used as coordinates in an n-dimensional space, so that a multiplication by the nxn matrics is a 'coordinate transformation', the derivative of ψ  plus Vxψ to form the side of Schrodinger equation is seen as a 'transformation of the coordinates of an Hilbert Space'.

So in the formation of quantum field theory, the dirac equation (now thought of as to describe the classic wave) morphs into what they call 'creation and anihillation operator'. Since when you differentiate a function, another function emerges, in the Schrodinger Equation for quantum harmonic oscillator, the derived function can be seen as 'the one describing one more particle. So the 'acting by the operator' is 'creating a particle' hence 'creation operator'. You can now see that it is just a mathematical operation, but the will mislead you into thinking that actual 'creation' is magically taking place!

So in turning the Dirac equation into 'creation and annihilation operators', we nolonger treat the ψ as a wave. Instead, it becomes parts of the 'constants' inside the operator, like the way 'V' is inside the Hamiltonian Operator. So we are left to wonder what it is that act like ψ in QFT. It turns out to be a 'wave of waves' termed 'wavefunctional'. From here, the picture of a 'wave of particle' is lost! So QFT simply ignores the question of what wave/particles are, and does something else, while insisting that it is innacurate to think of particles as in the classic QM where there is no 'creation' and 'annihilation'! Thus QFT actually contradicts QM!!